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(3)(a) i. Morishima, K. Endo, and T. Yonezawa, Chem. Phys. Len., 9, 203 (1971);(b) J. Amer. Chern. SOC.,93, 2048 (1971);(c) I. Morishima, T. Inubushi, K. Endo, T. Yonezawa, and K. Goto, ibid., 94,4812 (1972);(d) I. Morishima, K. Endo, and T. Yonezawa, J. Chem. Phys., 58, 3146 (1973). (4)R. Briere, H. Lemaire, and A. Rassat, Bui. SOC. Chim. Fr., 11, 3273 (1965). (5) P. L. Dahlstrom, J. L. Loo, and R. L. Harris, J. Magn. Resonance, in press.
(6)L. N. Mulay, "Magnetlc Susceptlbility," Interscience, New York, N.Y., 1963. (7)(a) F. L. Slejko, R. S. Drago, and D. G. Brown, J. Amer. Chem. SOC.,94, 9210 (1972);(b) T. D. Epley, Ph.D. Thesis, 1967, University of Illinois, Urbana, 111. 61801. (8) The general minimization routine, SIMPLEX, has been copyrighted by J. P. Chandler of the University of Indiana, 1965.
(9)R. W. Taft, E. Price, I. R. Fox, I. C. Lewis, K. K. Anderson, and G. T. Davis, J. Amer. Chem. SOC., 85,3146 (1963). (IO)(a) 0. N. La Mar and E. 0. Sherman, J. Amer. Chem. SOC., 92, 2691 (1970);(b) M. F. Rettig and R. S.Drago, ibid., 88, 2966 (1966);(c) W. D. Perry and R. S.Drago, bid., 93, 2183 (1971). (1 1) D. A. Deranleau, J. Amer. Chem. SOC., 91,4044 (1969). (12)(a) G.Egloff, "Physical Constants of Hydrocarbons," Vol. 2,ACS Monograph Series, Reinhold, New York, N.Y., 1940; (b) J. Timmermans, "Physico-Chemical Constants of Pure Organic Compounds," Elsevier, New York, N.Y., 1950. (13)Y. Murata and N. Mataga, Bull. Chem. SOC.Jap., 44,354 (1971). (14)C. J. Jameson and H. S. Gutowsky, J. Chem. Phys., 51,2790(1969). (15)R. Wiley and S.I. Miller, J. Amer. Chem. SOC., 94,3287 (1972). (16)N. Muller and R. C. Reiter, J. Chem. Phys., 42,3265 (1965). (17)I. A. Zlochower, W. R. Miller, Jr., and G. K. Fraenkel, J. Chem. Phys.,
42,3339 (1965).
Computation of the Intermolecular Vibrational Modes of a Tetrahedral Water Pentamer at the Core of an Ice-Like Water Cluster' Lester L. Shipman2*and Harold A. Scheraga" 2b Department of Chemistry, Cornel/ U6iversity, Ithaca, New York 14853 (Received May 23, 1974
The Shipman-Scheraga intermolecular potential energy function for water (SS potential) has been used to calculate the intermolecular vibrational modes of a tetrahedral water pentamer at the core of an ice-like water 17-mer. The calculated intermolecular vibrational spectrum was compared to the experimental infrared and inelastic neutron scattering spectra of ice Ih. Comparison of the vibrational frequencies of the pentamer at the core of the 17-mer with the vibrational frequencies of a single water molecule at the core of a tetrahedral pentamer water cluster indicates that vibrational coupling among the central water molecule and its four nearest neighbors increases the width of the librational and hindered translational bands, while leaving the mean librational and hindered translational frequencies almost unchanged. Integrated infrared absorptivities have been calculated for each intermolecular normal vibrational mode of the pentamer, and these integrated absorptivities have been summed to give the total integrated absorptivities for the librational and translational bands, for three different models for the calculation of the total dipole moment of the 17-mer. The calculated integrated infrared absorptivities for the hindered translational and librational bands were compared to the corresponding experimental values for ice Ih.
I. Introduction The intermolecular vibrational spectrum of ice Ih (ordinary ice) has been investigated experimentally by the infrared,3-14 neutron scattering,15-29 and Raman3,30-35spectroscopic techniques. In addition, there have been several theoretical s t ~ d i e s ~of~these * ~ intermolecular ~ ~ ~ ~ - ~ ~vibrations. While most theoretical st~dies27$29,36-~3 have used harmonic force constant models (0 0 0 bending, 0 0 stretch, etc.), the present study made use of the recently derived45 Shipman-Scheraga six-dimensional pair potential for water (SS potential). Intermolecular vibrational frequencies have been calculated from the multidimensional intermolecular potential in a manner similar to that of several previous s t ~ d i e s . ~The ~ , ~aims ~?~ of ~the present study were to investigate the effects of vibrational coupling among the central water molecule and its four nearest neighbors in an ice Ih arrangement and to compare several models for the induced dipoles in water in condensed phases. It was not the aim of our study to calculate the detailed
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The Journal of Physical Chemistry, Vol. 79, No. 4 , 1975
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intermolecular vibrational spectrum of ice Ih; such a calculation would require a consideration of the dynamics of many water molecules in a large portion of the ice lattice in order to properly account for long-range coupling; our aim was to obtain an understanding a t the molecular level of the effects of short-range coupling on the intermolecular vibrations of water molecules in ice-like environments. The model used for the study48 was a 17-mer consisting of a central water molecule, its 4 nearest neighbors, and its 12 next-to-nearest neighbors,49 with the outer 1 2 molecules held fixed and the inner core of 5 molecules allowed to undergo intermolecular vibrations. Intramolecular vibrational modes were not considered, and the structure of each monomer was held fixed a t the experimental infrared-microwave structure50 of the water monomer. The proton arrangement of the T'l-mer was that of the smallest hexagonal unit C36v (P63cm), for proton-ordered ice Ih. In real ice I h , the protons are known to be disordered on the basis of entropic consideration^^^ and neutron diffraction evidence.53 This ordered proton arrangement was used (or
Intermolecular Vibrational Modes of Water Pentamer all calculations in the present study; Prask, et a1.,Z9 have found that proton-ordered and proton-disordered ice give essentially the same vibrational spectrum. The SS potential was used to calculate the intermolecular normal vibrational modes of the central water molecule and its four nearest neighbors in the field of its 12 fixed next-to-nearest neighbors. The integrated infrared absorptivity was calculated for each of the intermolecular normal vibrational modes of the pentamer under the assumption that the total dipole moment of the 17-mer arose by three different mechanisms, uiz., permanent dipoles only, permanent plus molecule-induced dipoles, and permanent plus atom-induced dipoles (see section I11 for a discussion of these three models).
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TABLE I: Normal Mode Frequencies and Integrated Absorptivities of a Tetrahedral Pentamer in a n Ice-Like Environment
--__
w,
Modes Librational
10-6JK(w)
dw,icmg2
cm-'
a
b
C
830 803 792 728 718 715 705 690 690 689 666 656 641 636 634
0.334 0.002 0.240 0.315 0.290 0.098 0.000 0.114 0.074 0.033 0.032 0.112 0.028 0.001 0.017
0.320 0.003 0.217 0.318 0.304 0.093 0.000 0.112 0.072 0.032 0.032 0.118 0.033 0.001 0.020
0.186 0.002 0.131 0.200 0.208 0.057 0.000 0.077 0.046 0.024 0.019 0.067 0.017 0.000 0.010
11.Calculation of t h e Intermolecular Normal Vibrational Frequencies The structure of the 17-mer for which the normal vibrational frequencies were calculated was obtained by minimizing the empirical SS intermolecular potential energy45 with respect to the 30 intermolecular degrees of freedom of the tetrahedral water pentamer at the core of the 17-mer, starting from the proton-ordered ice I h structure with an 0.001 0.003 0.005 Hindered 268 0 0 distance of 2.7493 fi. This particular distance was translational 262 0.000 0.004 0.009 chosen previously45 as an estimate of the 0 0 distance 258 0.002 0.007 0.012 in real ice I h at O O K . The outer 1 2 molecules of the 17-mer 208 0.002 0.006 0.010 were held fixed for all calculations. The intermolecular de202 0.000 0.002 0.006 grees of freedom were rotations about the three principal 0.000 199 0.002 0.008 axes for the moments of inertia and translations of the cen186 0.002 0.007 0.011 ter of mass along three space-fixed Cartesian axes (a total of 184 0.000 0.004 0.011 six degrees of freedom per water) for the central water and 182 0.000 0.003 0.006 each of its four nearest neighbors (a total of 30 degrees of 180 0.000 0.001 0.001 freedom). The five molecules (tetrahedral pentamer) that 179 0.001 0.004 0.007 were allowed to rotate and translate during the minimiza0.000 169 0.001 0.004 tion translated and rotated only very slightly from starting 100 0.000 0.000 0.000 to final positions (the average absolute translation along 83 0.000 0.002 0.002 the Cartesian axes was 0.03 fi and the average absolute 80 0,001 0.003 0.004 rotation about the principal axes was 2'). a Permanent dipoles only. Permanent plus molecule-induced dipoles. Permanent plus atom-induceddipoles. The harmonic force constant matrix (mass included) was calculated numerically at the minimum-potential-energy structure, and subsequent diagonalization of this matrix There is general agreement among the inelastic neutron gave the normal modes (eigenvectors of the force constant scattering s t u d i e ~ l on ~ - ice ~ ~ I h that the hindered translamatrix) and normal vibrational frequencies [1/(2nc) times tional band extends from quite low frequencies (
